In reply to Aeneas:
Honestly, they're both relics of the past that make life unnecessarily hard for yourself. You'll find it much easier to pick up ice climbing with any good leashless tool than one with leashes, and physically its actually easier this way once the ice gets steeper. Its also more fun than being strapped in!
Go for some DMM Rebels if you're after a budget tool, although you have to be aware that these really don't swing as well and gain as secure placements as the higher end of tools on the market. The next step up is the Black Diamond Vipers or Petzl Quarks if you can't get these (not quite as good).
The next step up is the Nomics, which are my axes of choice for hard alpine climbing thanks to their more radical curve which works better on mixed ground and the fact that they are hands down the best axe for steep ice. They don't work perfectly in some types of snow, but still do work well enough in my opinion.
I hope that answers your question! Also don't forget that crampon choice is very important as well; you'll find that having horizontal rather than vertical front points makes a huge improvement to your climbing.